Download Sum of Angles in a Polygon

InterMath
Title: Sum of Angles in a Polygon
Problem Statement
What is the sum of the angles of a triangle? of a quadrilateral? of a pentagon? of a hexagon?
What is the sum of the angles in convex polygons in terms of the number of sides? Move the
mouse pointer over the picture for a hint.
Problem setup
I am trying to find the sum of a triangle’s angle in a triangle, quadrilateral, and a pentagon. I am
also trying to determine the sum of the angles in a convex polygon. According to the dictionary
from the Intermath website, a convex polygon is a polygon that has all of its interior angles less
than 180.
M6G1
M6P1
M6P3
M6P4
Students will further develop their understanding of plane and solid geometric figures, incorporating the
use of appropriate technology and using this knowledge to solve authentic problems.
Students will further develop their understanding of plane figures.
a. Determine and use lines of symmetry.
b. Investigate rotational symmetry, including degree of rotation.
c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar
plane figures.
d. Interpret and sketch simple scale drawings.
e. Solve problems involving scale drawings.
Students will solve problems (using appropriate technology). a. Build new mathematical knowledge
through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
Use the language of mathematics to express mathematical ideas precisely.
Students will make connections among mathematical ideas and to other disciplines. a. Recognize and
use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent
whole.
Plans to Solve/Investigate the Problem
I plan to construct a quadrilateral, pentagon, and a hexagon in GSP. Next, I plan to create as
many triangles as possible by drawing a diagonal from a single vertex to other vertices in each
polygon. I’m constructing triangles because I know that the sum of the measure of the interior
angles of a triangle measure 180°. Then I plan to measure the interior angles of each polygon to
determine the sum of the angles in an individual polygon.
Investigation/Exploration of the Problem
I started my investigation by constructing a triangle in GSP. Then I measured the angles of the
triangle. The sum of the angles totaled 180.
D
N
C
tri angl e
no dia gonal
L
K
E
hexagon
three di agonal s
four tria ngles
F
O pentagon
two d iagIona ls
three tri angles
R
quadi l ateral
one di agonal
two triangl es
Q
A
B
P
M
J
H
G
Next, I constructed a quadrilateral. After I constructed the quadrilateral, I used one of the
vertices and drew a diagonal to all the angles that are not adjacent to that vertex. I noticed that
this diagonal created two triangles in the quadrilateral, three diagonals in the pentagon, and four
diagonals in the hexagon. So I assume that the measure of the interior angles of the quadrilateral
would be 2(180) or 360°, pentagon 3(180) or 540°, hexagon 4(180) or 720° and a decagon
8(180) or 1460°.
mCAB = 90.00
Triangle
mABC = 48.88
mBCA = 41.12
C
Trian gle ABC
A
mBCA+mABC+mCAB = 180.00
B
Quadrilateral:
A quadrilateral can be divided in to two triangles which has an angle measure of 180° each,
hence 180*2 or 360°.
Rectangl e DEFG
E
G
mFDE = 89.50
mDEG = 90.50
mEGF = 89.80
mGFD = 90.21
D
F
mFDE+mDEG+mEGF+mGFD
= 360.00
Pentagon
A pentagon can be divided into three distinct triangles. Therefore, the sum of the angles of a
pentagon is 3*180 or 540°, so there seem to be a pattern forming. The pattern is the number of
sides of a polygon minus two will give you the number of triangles in a polygon.
A pentagon has five sides. I subtracted 5-2=3. There are three distinct triangles in a pentagon.
mLKO = 103.73
K
L
mMLK = 126.22
M
mNML = 92.47
O
mKON = 130.20
N
mONM = 87.39
mMLK+mLKO+mKON+mONM+mNML = 540.00
Hexagon
A Hexagon has three diagonals drawn from a single vertex. These diagonals form four triangles;
therefore the sum of the angles is 4*180 or 720°.
M
N
mNMR = 117.63
mONM = 124.23
mPON = 119.40
mQPO = 116.37
mRQP = 117.65
O
mQRM = 124.73
R
Q
P
mNMR+mONM+mPON+mQPO+mRQP+mQRM = 720.00
I noticed that a relationship between the sum of angles in a polygon can be determine by the
number of diagonals from a single vertex, and the number of diagonals that can be created from a
single vertex can be determined by subtracting three from the number of sides of a polygon.
After subtracting three from the number of sides of a polygon, the remaining sides will give us
the number of triangles in a polygon. Then the number of triangles can be multiplied the by 180.
Example: A decagon has 10 sides, so I subtracted 3 from 10. 10-3 gives me 7 diagonals that can
be created in a decagon. Seven diagonals create 8 triangles, which will be 8*180, or 1440°. The
number of triangles can also be determined by subtracting 2 from the number of sides and
multiplying the remaining sides by 180. 10-2=8 this is shown below in the decagon. This
equation will give you the sum of the angles of any polygon.
A
mJAB = 143.57
J
B
mABC = 143.81
mBCD = 144.50
mCDE = 142.11
mDEF = 144.53
mEFG = 146.89
I
C
mFGH = 141.34
mGHI = 146.06
mHIJ = 142.11
mIJA = 145.08
D
H
I multip li ed the number of tri angle s insi de the decagon
times 180 degrees
1808 = 1440.00
G
E
F
mJAB+mABC+mBCD+mCDE+mDEF+mEFG+mFGH+mGHI+mHIJ+mIJA = 1440.00
Extensions of the Problem
As an extension of this investigation I would have my students created a chart of the number of
diagonals and sum of the angle measurements. This chart would also include an equation that
would work for any polygon.
Polygons
Number of sides
Number of diagonals
drawn from a single vertex
Sum of the Angle measure (N2)*180
Triangle
3
0
1*180=180
Quadrilateral
4
1
2*180=360
Pentagon
5
2
3*180=540
Hexagon
6
3
4*180=720
Heptagon
7
4
5*180=900
Octagon
8
5
6*180=1080
N-gon
N
n-3
(n-2)*180
.
Author & Contact
Norma Smith and contact information.
[email protected]
Link(s) to resources, references, lesson plans, and/or other materials
http://www.geom.uiuc.edu/~dwiggins/conj07.html
http://mathworld.wolfram.com/RegularPolygon.html